A Periodic Table for Black Hole Orbits Janna Levin Dynamical Systems approach Gabe Perez-Giz NSF Fellow Becky Grossman NSF Fellow

Gravitational Waves are a fundamental prediction of General Relativity Something changes in the universe -- a distant object moves -- and we don’t know about it until a gravitational wave traveling at the speed of light makes it to us to communicate that change Inspired by the imminent Advanced LIGO and the proposed space mission LISA Gravitational Wave Detection Relies on a Knowledge Dynamics

Binary Black Holes (BBH) Many Varieties 3. Intermediate Mass BBH 2. Stellar Mass BBH formed by tidal capture in globular clusters 1. Stellar Mass BBH from a stellar binary 4. Supermassive Black Holes with a compact companion

2body problem is insoluble Spectrum of orbits Newtonian elliptical orbits Relativistic 2body with spins - chaotic Relativistic precession In the strong-field regime periodic table of BH orbits

Might expect orbits of this kind Circular orbit Nearby low-eccentricity precessing orbit Highly-eccentric orbit unbound

Instead, in strong-field zoom-whirl orbits prevalent Multi-leaf clovers zooms whirls Extreme form of perihelion precession. Doesn’t close.

Anatomy of zoom-whirl orbits zooms whirls

We build a periodic table of black hole orbits Between the ground state -- the stable circular orbit And the energy of ionization -- the escape orbit (or merger) Lies an infinite set of periodic orbits All generic orbits are defined by this periodic set

Spinning Black Holes Kerr Spacetime In the 30+ years since Carter found the solutions, the periodic spectrum was overlooked Carter found the geodesics around a Kerr black hole We know orbits around a spinning black hole

Taxonomy of Periodic Orbits Kerr, Equatorial Every orbit is defined by a rational number z = 3 w = 1 v = 1 Look at one radial cycle

Taxonomy of Periodic Orbits Kerr, Equatorial Every orbit is defined by a rational number z = 3 w = 1 v = 2 And this is a unique 3 leaf orbit

Finding periodic orbits with rational numbers Physically, the rational corresponds to the accumulated angle in one radial cycle

Finding periodic orbits with rational numbers Physically, the rational corresponds to the accumulated angle in one radial cycle

Another derivation of q Every equatorial orbit has two fundamental frequencies Radial frequency; is the radial period Angular frequency; average of over For periodic orbits, their ratio is rationally related and are the same frequencies derived from an action-angle formulation of the dynamics

Are all periodic orbits allowed? No, as illustrated in this energy-level diagram Highest Energy (unstable circular orbit) Lowest Energy (stable circular orbit) q >1+1/3

No Mercury-like orbits in the strong field q > 1+1/3 there are no tightly precessing ellipses since q for these would be near zero All orbits in the strong-field exhibit zoom and/or whirl behavior

A periodic table of black hole orbits Energy increases up to down and left to right Columns are w-bands Rows are v/z

All orbits, ALL, are defined by the periodic set High z orbits as perturbations of low z orbits (3,1, 1) q=1+1/3 (300,1, 103) q=1+103/300 Perturb around closed orbit to get precession

We can apply the periodic taxonomy to any description of binary black holes No known analytic solution to the metric or orbits. Comparable mass black hole pairs

Not enough constants of motion Unless restrict to SO only

Fully three-dimensional motion Understanding these requires some new machinery Perihelion precession + Spin precession

Fully 3D motion now since spin precession drags out of the plane

Fully three-dimensional motion as 2d orbital motion + precession of plane the orbit looks simple in the orbital plane

Suggests a way to find I.C. for Numerical Relativity Test PN expansion Compute GWs Find closed orbits in orbital plane

A periodic Table in the Orbital Plane

These orbital plane periodics are generally not closed in 3D We have not required a rational in 

Not necessary to discuss fully closed 1.These are a subset of the set of periodics in the orbital plane. Since the subset is dense, our set of periodics must be dense 2. Since q  and  are related for fixed L, an orbit that is near a given orbital periodic is near it in the full 3D

Not necessary to discuss fully closed

Fully closed in three-dimensions A fully closed orbit after one full orbit A fully closed orbit after two full orbits

Utility of Rational Spectrum Gravitational wave astronomy Computation of Inspiral Transition to chaos with higher-order spin All eccentric orbits will show zooms and all high spin show whirls Data analysis (template of periodic orbits) Inspiral through periodic orbits Fourier decomposition in 1 frequency instead of 3

The famed ISCO is a homoclinic orbit with zero e An orbit with any residual e will transition to plunge through one of these separatrices Separatrix from Inspiral to Plunge

The homoclinic orbit in phase space The divide between bound and plunging, whirling and not-whirling, chaotic and not chaotic

The periodic orbits To pack a proliferating infinity of orbits in a finite phase space form a Fractal Homoclinic tangle Chaos develops around certain unstable periodic orbits

What don’t we Know If the companion spins - orbits are strongly altered If there are two black holes of comparable mass There are no known exact solutions to the orbits of these BH binaries

Fractal Basin Boundaries z x y look at initial basins as the spin angles are varied find they’re fractal Post-Newtonian approximation to the 2body problem Black merger White stable Grey escape

Fractal Basin Boundary

Chaotic scattering off an a fractal set of periodic orbits

In Brief First test of GR the precession of the perhelion of mercury Almost a century later, looking for precession of multi-leaf orbits The precessions will shape the gravitational waves